A spline method for solving fourth order singularly perturbed boundary value problem

Abstract

In this paper, singularly perturbed boundary value problem of fourth order ordinary differential equation with a small positive parameter multiplying with the highest derivative of the form$$\begin{gathered}\varepsilon u^{(4)}(x)+p(x) u^{\prime \prime}(x)+q(x) u(x)=r(x), 0 \leq x \leq 1, \\u(0)=\gamma_0, u(1)=\gamma_1, u^{\prime \prime}(0)=\eta_0, u^{\prime \prime}(1)=\eta_1, 0 \leq \varepsilon \leq 1\end{gathered}$$is considered. We have developed a numerical technique for the above problem using parametric and polynomial septic spline method. The method is shown to have second and fourth order convergent depending on the choice of parameters involved in the method. Truncation error and boundary equations are obtained. The method is tested on an example and the results are found to be in agreement with the theoretical analysis.